Encyclopedia of Microtonal Music Theory

schisma / skhisma

[Joe Monzo]

1. first definition, by Philolaus

Historical priority of usage of the term "schisma" is by Philolaus (fl. c. 400s BC), as quoted by Boethius, to refer to an interval that is considerably larger than the schisma we normally mean today (see below #2 for that).

Philolaus described the schisma as an integral logarithmic 1/2 of the comma (which today we call the pythagorean-comma):

2. usual modern sense, ~2 cents

In its usual modern sense, a term coined by Alexander Ellis in his translation of Helmholtz, On the Sensations of Tone, and originally spelled skhisma. It designates an extremely small interval, just barely discernible to human pitch-detection.

Unqualified, it is the difference between the 51 just "major 3rd" and the 3-8 pythagorean "diminished 4th", and has an interval size of approximately 1/50 Semitone [= ~ 2 cents]:

Tuning treatises before c.1970 often defined the skhisma as the 887:886 ratio without comment on the fact of its being an approximation, particularly in the German literature of c.1850-1950. A good example is in Helmholtz, On the Sensations of Tone....

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3.with qualifier, other small interval

As no combination of different prime numbers will ever produce ratios which have exactly the same interval size, if cycles of a particular ratio are calculated far enough, very small intervals like this eventually appear between ratios having different sets of prime factors. When these are under consideration, the term "schisma" is qualified with a latin word designating the higher prime, with the assumption that the other prime being compared is a more familiar one, almost always 3.

Thus, we get the "septimal schisma", which is the difference between the 3-14 pythagorean "doubly diminished 8ve" and the 71 harmonic "minor 7th", and has an interval size of approximately 1/26 Semitone:

Likewise, there is the "nondecimal schisma", which is the difference between the 191 harmonic "augmented 2nd" and the 'standard' 3-3 pythagorean "minor 3rd", and which has an interval size of approximately 1/30 Semitone: